# A Numerical Study on Impacts of Sediment Erosion/Deposition on Debris Flow Propagation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulations

#### 2.1. Numerical Models That Will Be Used in This Paper

#### 2.2. Governing Equations

^{2}is gravitational acceleration $,{\rho}_{\mathit{o}}$ = ${\rho}_{\mathit{w}}$p + ${\rho}_{\mathit{s}}$(1 − p) is the density of the water-saturated bed, ${\mathit{r}}_{\mathit{w}}$ is the ratio of the upper-interface resistance to the bed resistance, ${\mathit{e}}_{\mathit{w}}$ is the entrainment from the ambient fluid layer, p is the bed sediment porosity, ${\tau}_{\mathit{b}\mathit{x}}\mathrm{and}{\tau}_{by}$ are the bed shear stresses in x and y directions, respectively; ${\mathit{c}}_{\mathit{D}}$ is the drag coefficient, E is the sediment entrainment, and D is the deposition flux, which arises from the sediment exchange with the bed, while $-\frac{\partial \mathit{z}}{\partial \mathit{x}}$ and $-\frac{\partial \mathit{z}}{\partial \mathit{y}}$ are the bed slopes in x and y directions, respectively.

#### 2.3. Empirical Relations

_{b}c and E = wE

_{s};

_{s}is entrainment coefficient; ${\mathit{r}}_{\mathit{b}}$ is the difference between the layer-averaged concentration and the near-bed concentration; and ${\mathit{u}}_{*}=\sqrt{{\mathit{c}}_{\mathit{D}}\mathit{u}\overline{\mathit{U}}}$ and ${\mathit{v}}_{*}=\sqrt{{\mathit{c}}_{\mathit{D}}\mathit{v}\overline{\mathit{U}}}$ are the bed shear velocities in x and y directions.

#### 2.4. Numerical Method

**U**is the vector of conserved variables,

**F**and

**G**are the advection terms,

**S**

_{b}is the vector of bed source term, and

**S**

_{f}is the vector of other source terms, including entrainment, bed deformation effects, and friction terms.

**U**, and the superscripts R and L represent the right and left sides of the interface of the two cells. For the φ, the MinBee limiter function was used [45]. Vector r was solved as follows:

_{lim}is the critical Richardson number. Ri

_{lim}= 1.0 was used in this study.

## 3. Model Validation

#### 3.1. Experimental Data Description

#### 3.2. Verification with One-Dimensional Laboratory Test (Subaerial)

#### 3.3. Verification with One-Dimensional Laboratory Test

## 4. Model Application

#### 4.1. Application to Debris Flow over a Continental Shelf that Leads to Deep Sea

#### 4.2. Sensitivity Analyses

#### 4.2.1. Impact of Sediment Size and Bed Slope Angle on Bed Thickness

#### 4.2.2. Impact of Bed Porosity and Debris Concentration on Bed Thickness

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Comparison of debris-flow height between experimental, analytical, and numerical data sets for case WK87 (“Exp.” data from Wright and Krone [34]; “Analytic” data from Huang and García [8]; IM from Imran et al. [3]; Dong from Dong et al. [46]; “Num. QD19, QD20” data from Qian and Das [7], Qian et al. [1], resp.).

**Figure 2.**(

**a**) Sensitivity result for different concentrations using TC and TCQD and comparison of debris-flow height with experimental and numerical data sets for case WK87 (“Exp.” data from Wright and Krone [34]). (

**b**) Velocity profiles for QDnew, TC, and TCQD.

**Figure 6.**Debris heights along the slope for QDnew, TC, and TCQD at different times. (

**a**–

**d**) show the debris flow heights at 5 s, 100 s, 500 s, and 7200 s, respectively.

**Figure 7.**Velocity profiles of QDnew, TCQD, and TC at different times. (

**a**–

**d**) show the velocity profiles at 5 s, 100 s, 500 s, and 7200 s, respectively.

**Figure 8.**Concentration profiles of TCQD and TC at different times. (

**a**–

**d**) show the concentration profiles at 5 s, 100 s, 500 s, and 7200 s, respectively.

**Figure 9.**Bed deformation profiles of TCQD and TC at different times. (

**a**–

**d**) show the bed deformation profiles at 5 s, 100 s, 500 s, and 7200 s, respectively.

**Figure 10.**(

**d**) Comparisons of the evolution of the sediment bed. (

**a**) different sediment sizes (

**b**) different bed slopes, (

**c**) different slopes, and (

**d**) different concentrations.

Cases | Models | Estimation (%) |
---|---|---|

WK87 | QDnew | 7.9 |

TCQD | −2.4 | |

TC | 29.3 | |

M2a | QDnew | 1.48 |

TCQD | 3.06 | |

TC | 38.7 | |

M3a | QDnew | −4.2 |

TCQD | −2.4 | |

TC | 12.9 |

Case No. | d (µm) | ${\mathit{S}}_{\mathit{b}}\mathbf{Angle}(\xb0)$ | Porosity | c | ${\mathit{\tau}}_{\mathit{y}\mathit{i}\mathit{e}\mathit{l}\mathit{d}}\left(\mathbf{Pa}\right)$ | $\mathit{\mu}$ (Pa.s) | Remarks |
---|---|---|---|---|---|---|---|

1 | 37 | 6 | 0.5 | 0.27 | 100 | 20 | Reference case |

2 | 50 | 6 | 0.5 | 0.27 | 100 | 20 | Impact of sediment size |

3 | 100 | 6 | 0.5 | 0.27 | 100 | 20 | |

4 | 37 | 3 | 0.5 | 0.27 | 100 | 20 | Impact of bed slope angle |

5 | 37 | 9 | 0.5 | 0.27 | 100 | 20 | |

6 | 37 | 6 | 0.2 | 0.27 | 100 | 20 | Impact of bed porosity |

7 | 37 | 6 | 0.7 | 0.27 | 100 | 20 | |

8 | 37 | 6 | 0.5 | 0.5 | 100 | 20 | Impact of concentration/mass/density |

9 | 37 | 6 | 0.5 | 0.8 | 100 | 20 |

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**MDPI and ACS Style**

Adebiyi, A.A.; Hu, P.
A Numerical Study on Impacts of Sediment Erosion/Deposition on Debris Flow Propagation. *Water* **2021**, *13*, 1698.
https://doi.org/10.3390/w13121698

**AMA Style**

Adebiyi AA, Hu P.
A Numerical Study on Impacts of Sediment Erosion/Deposition on Debris Flow Propagation. *Water*. 2021; 13(12):1698.
https://doi.org/10.3390/w13121698

**Chicago/Turabian Style**

Adebiyi, Abiola Abraham, and Peng Hu.
2021. "A Numerical Study on Impacts of Sediment Erosion/Deposition on Debris Flow Propagation" *Water* 13, no. 12: 1698.
https://doi.org/10.3390/w13121698